3.23.29 \(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx\) [2229]

Optimal. Leaf size=256 \[ -\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {b^{3/2} (7 b B d-2 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}} \]

[Out]

-2/5*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(5/2)-2/15*(-2*A*b*e-5*B*a*e+7*B*b*d)*(b*x+a)^(5/2)/e^2/(-a
*e+b*d)/(e*x+d)^(3/2)-b^(3/2)*(-2*A*b*e-5*B*a*e+7*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/
e^(9/2)-2/3*b*(-2*A*b*e-5*B*a*e+7*B*b*d)*(b*x+a)^(3/2)/e^3/(-a*e+b*d)/(e*x+d)^(1/2)+b^2*(-2*A*b*e-5*B*a*e+7*B*
b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/e^4/(-a*e+b*d)

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Rubi [A]
time = 0.12, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 49, 52, 65, 223, 212} \begin {gather*} -\frac {b^{3/2} (-5 a B e-2 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}+\frac {b^2 \sqrt {a+b x} \sqrt {d+e x} (-5 a B e-2 A b e+7 b B d)}{e^4 (b d-a e)}-\frac {2 b (a+b x)^{3/2} (-5 a B e-2 A b e+7 b B d)}{3 e^3 \sqrt {d+e x} (b d-a e)}-\frac {2 (a+b x)^{5/2} (-5 a B e-2 A b e+7 b B d)}{15 e^2 (d+e x)^{3/2} (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) - (2*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(a + b*x
)^(5/2))/(15*e^2*(b*d - a*e)*(d + e*x)^(3/2)) - (2*b*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(a + b*x)^(3/2))/(3*e^3*(b*
d - a*e)*Sqrt[d + e*x]) + (b^2*(7*b*B*d - 2*A*b*e - 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(e^4*(b*d - a*e)) -
(b^(3/2)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/e^(9/2)

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{7/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {(7 b B d-2 A b e-5 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{5/2}} \, dx}{5 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}+\frac {(b (7 b B d-2 A b e-5 a B e)) \int \frac {(a+b x)^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e^2 (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {\left (b^2 (7 b B d-2 A b e-5 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{e^3 (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {\left (b^2 (7 b B d-2 A b e-5 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{2 e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {(b (7 b B d-2 A b e-5 a B e)) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {(b (7 b B d-2 A b e-5 a B e)) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{5 e (b d-a e) (d+e x)^{5/2}}-\frac {2 (7 b B d-2 A b e-5 a B e) (a+b x)^{5/2}}{15 e^2 (b d-a e) (d+e x)^{3/2}}-\frac {2 b (7 b B d-2 A b e-5 a B e) (a+b x)^{3/2}}{3 e^3 (b d-a e) \sqrt {d+e x}}+\frac {b^2 (7 b B d-2 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^4 (b d-a e)}-\frac {b^{3/2} (7 b B d-2 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 205, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {a+b x} \left (2 a^2 e^2 (2 B d+3 A e+5 B e x)+2 a b e \left (A e (5 d+11 e x)+B \left (20 d^2+49 d e x+35 e^2 x^2\right )\right )+b^2 \left (2 A e \left (15 d^2+35 d e x+23 e^2 x^2\right )-B \left (105 d^3+245 d^2 e x+161 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{15 e^4 (d+e x)^{5/2}}+\frac {b^{3/2} (-7 b B d+2 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{e^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

-1/15*(Sqrt[a + b*x]*(2*a^2*e^2*(2*B*d + 3*A*e + 5*B*e*x) + 2*a*b*e*(A*e*(5*d + 11*e*x) + B*(20*d^2 + 49*d*e*x
+ 35*e^2*x^2)) + b^2*(2*A*e*(15*d^2 + 35*d*e*x + 23*e^2*x^2) - B*(105*d^3 + 245*d^2*e*x + 161*d*e^2*x^2 + 15*
e^3*x^3))))/(e^4*(d + e*x)^(5/2)) + (b^(3/2)*(-7*b*B*d + 2*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(S
qrt[e]*Sqrt[a + b*x])])/e^(9/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1091\) vs. \(2(222)=444\).
time = 0.09, size = 1092, normalized size = 4.27

method	result	size
default	\(\frac {\left (90 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{2} e^{2} x +30 B \,b^{2} e^{3} x^{3} \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-60 A \,b^{2} d^{2} e \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}+90 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d \,e^{3} x^{2}+75 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} e^{4} x^{3}-105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d \,e^{3} x^{3}-315 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{2} e^{2} x^{2}-92 A \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-315 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{3} e x +75 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d^{3} e -20 B \,a^{2} e^{3} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-8 B \,a^{2} d \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-105 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{4}-140 A \,b^{2} d \,e^{2} x \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-20 A a b d \,e^{2} \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}-12 A \,a^{2} e^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+210 B \,b^{2} d^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-196 B a b d \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+30 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} d^{3} e +225 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d \,e^{3} x^{2}+225 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{2} d^{2} e^{2} x -140 B a b \,e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+322 B \,b^{2} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-44 A a b \,e^{3} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+490 B \,b^{2} d^{2} e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-80 B a b \,d^{2} e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+30 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{3} e^{4} x^{3}\right ) \sqrt {b x +a}}{30 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, \left (e x +d \right )^{\frac {5}{2}} e^{4}}\)	\(1092\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/30*(90*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^2*e^2*x+30*B*b^2*
e^3*x^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-60*A*b^2*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+90*A*ln(1/2*(2*
b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d*e^3*x^2+75*B*ln(1/2*(2*b*e*x+2*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*e^4*x^3-105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/
2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d*e^3*x^3-315*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
+a*e+b*d)/(b*e)^(1/2))*b^3*d^2*e^2*x^2-92*A*b^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-315*B*ln(1/2*(2*b*
e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^3*e*x+75*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*d^3*e-20*B*a^2*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-
8*B*a^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+
a*e+b*d)/(b*e)^(1/2))*b^3*d^4-140*A*b^2*d*e^2*x*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-20*A*a*b*d*e^2*(b*e)^(1/2)
*((b*x+a)*(e*x+d))^(1/2)-12*A*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*B*b^2*d^3*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)-196*B*a*b*d*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+30*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(
1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^3*e+225*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a
*e+b*d)/(b*e)^(1/2))*a*b^2*d*e^3*x^2+225*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e
)^(1/2))*a*b^2*d^2*e^2*x-140*B*a*b*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+322*B*b^2*d*e^2*x^2*((b*x+a)*(e
*x+d))^(1/2)*(b*e)^(1/2)-44*A*a*b*e^3*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+490*B*b^2*d^2*e*x*((b*x+a)*(e*x+d)
)^(1/2)*(b*e)^(1/2)-80*B*a*b*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+30*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d)
)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*e^4*x^3)*(b*x+a)^(1/2)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+
d)^(5/2)/e^4

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 4.87, size = 798, normalized size = 3.12 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{2} d^{4} - {\left (5 \, B a b + 2 \, A b^{2}\right )} x^{3} e^{4} + {\left (7 \, B b^{2} d x^{3} - 3 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d x^{2}\right )} e^{3} + 3 \, {\left (7 \, B b^{2} d^{2} x^{2} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} x\right )} e^{2} + {\left (21 \, B b^{2} d^{3} x - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3}\right )} e\right )} \sqrt {b} e^{\left (-\frac {1}{2}\right )} \log \left (b^{2} d^{2} + 4 \, {\left (b d e + {\left (2 \, b x + a\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\left (-\frac {1}{2}\right )} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) - 4 \, {\left (105 \, B b^{2} d^{3} + {\left (15 \, B b^{2} x^{3} - 6 \, A a^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} x\right )} e^{3} + {\left (161 \, B b^{2} d x^{2} - 14 \, {\left (7 \, B a b + 5 \, A b^{2}\right )} d x - 2 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d\right )} e^{2} + 5 \, {\left (49 \, B b^{2} d^{2} x - 2 \, {\left (4 \, B a b + 3 \, A b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{60 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} e^{4}\right )}}, \frac {15 \, {\left (7 \, B b^{2} d^{4} - {\left (5 \, B a b + 2 \, A b^{2}\right )} x^{3} e^{4} + {\left (7 \, B b^{2} d x^{3} - 3 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d x^{2}\right )} e^{3} + 3 \, {\left (7 \, B b^{2} d^{2} x^{2} - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} x\right )} e^{2} + {\left (21 \, B b^{2} d^{3} x - {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3}\right )} e\right )} \sqrt {-b e^{\left (-1\right )}} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {-b e^{\left (-1\right )}}}{2 \, {\left (b^{2} d x + a b d + {\left (b^{2} x^{2} + a b x\right )} e\right )}}\right ) + 2 \, {\left (105 \, B b^{2} d^{3} + {\left (15 \, B b^{2} x^{3} - 6 \, A a^{2} - 2 \, {\left (35 \, B a b + 23 \, A b^{2}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} + 11 \, A a b\right )} x\right )} e^{3} + {\left (161 \, B b^{2} d x^{2} - 14 \, {\left (7 \, B a b + 5 \, A b^{2}\right )} d x - 2 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d\right )} e^{2} + 5 \, {\left (49 \, B b^{2} d^{2} x - 2 \, {\left (4 \, B a b + 3 \, A b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{30 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} e^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

[-1/60*(15*(7*B*b^2*d^4 - (5*B*a*b + 2*A*b^2)*x^3*e^4 + (7*B*b^2*d*x^3 - 3*(5*B*a*b + 2*A*b^2)*d*x^2)*e^3 + 3*
(7*B*b^2*d^2*x^2 - (5*B*a*b + 2*A*b^2)*d^2*x)*e^2 + (21*B*b^2*d^3*x - (5*B*a*b + 2*A*b^2)*d^3)*e)*sqrt(b)*e^(-
1/2)*log(b^2*d^2 + 4*(b*d*e + (2*b*x + a)*e^2)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(b)*e^(-1/2) + (8*b^2*x^2 + 8*a
*b*x + a^2)*e^2 + 2*(4*b^2*d*x + 3*a*b*d)*e) - 4*(105*B*b^2*d^3 + (15*B*b^2*x^3 - 6*A*a^2 - 2*(35*B*a*b + 23*A
*b^2)*x^2 - 2*(5*B*a^2 + 11*A*a*b)*x)*e^3 + (161*B*b^2*d*x^2 - 14*(7*B*a*b + 5*A*b^2)*d*x - 2*(2*B*a^2 + 5*A*a
*b)*d)*e^2 + 5*(49*B*b^2*d^2*x - 2*(4*B*a*b + 3*A*b^2)*d^2)*e)*sqrt(b*x + a)*sqrt(x*e + d))/(x^3*e^7 + 3*d*x^2
*e^6 + 3*d^2*x*e^5 + d^3*e^4), 1/30*(15*(7*B*b^2*d^4 - (5*B*a*b + 2*A*b^2)*x^3*e^4 + (7*B*b^2*d*x^3 - 3*(5*B*a
*b + 2*A*b^2)*d*x^2)*e^3 + 3*(7*B*b^2*d^2*x^2 - (5*B*a*b + 2*A*b^2)*d^2*x)*e^2 + (21*B*b^2*d^3*x - (5*B*a*b +
2*A*b^2)*d^3)*e)*sqrt(-b*e^(-1))*arctan(1/2*(b*d + (2*b*x + a)*e)*sqrt(b*x + a)*sqrt(x*e + d)*sqrt(-b*e^(-1))/
(b^2*d*x + a*b*d + (b^2*x^2 + a*b*x)*e)) + 2*(105*B*b^2*d^3 + (15*B*b^2*x^3 - 6*A*a^2 - 2*(35*B*a*b + 23*A*b^2
)*x^2 - 2*(5*B*a^2 + 11*A*a*b)*x)*e^3 + (161*B*b^2*d*x^2 - 14*(7*B*a*b + 5*A*b^2)*d*x - 2*(2*B*a^2 + 5*A*a*b)*
d)*e^2 + 5*(49*B*b^2*d^2*x - 2*(4*B*a*b + 3*A*b^2)*d^2)*e)*sqrt(b*x + a)*sqrt(x*e + d))/(x^3*e^7 + 3*d*x^2*e^6
+ 3*d^2*x*e^5 + d^3*e^4)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (234) = 468\).
time = 1.93, size = 668, normalized size = 2.61 \begin {gather*} \frac {{\left (7 \, B b^{2} d {\left | b \right |} - 5 \, B a b {\left | b \right |} e - 2 \, A b^{2} {\left | b \right |} e\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \frac {{\left ({\left ({\left (b x + a\right )} {\left (\frac {15 \, {\left (B b^{9} d^{2} {\left | b \right |} e^{6} - 2 \, B a b^{8} d {\left | b \right |} e^{7} + B a^{2} b^{7} {\left | b \right |} e^{8}\right )} {\left (b x + a\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}} + \frac {23 \, {\left (7 \, B b^{10} d^{3} {\left | b \right |} e^{5} - 19 \, B a b^{9} d^{2} {\left | b \right |} e^{6} - 2 \, A b^{10} d^{2} {\left | b \right |} e^{6} + 17 \, B a^{2} b^{8} d {\left | b \right |} e^{7} + 4 \, A a b^{9} d {\left | b \right |} e^{7} - 5 \, B a^{3} b^{7} {\left | b \right |} e^{8} - 2 \, A a^{2} b^{8} {\left | b \right |} e^{8}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} + \frac {35 \, {\left (7 \, B b^{11} d^{4} {\left | b \right |} e^{4} - 26 \, B a b^{10} d^{3} {\left | b \right |} e^{5} - 2 \, A b^{11} d^{3} {\left | b \right |} e^{5} + 36 \, B a^{2} b^{9} d^{2} {\left | b \right |} e^{6} + 6 \, A a b^{10} d^{2} {\left | b \right |} e^{6} - 22 \, B a^{3} b^{8} d {\left | b \right |} e^{7} - 6 \, A a^{2} b^{9} d {\left | b \right |} e^{7} + 5 \, B a^{4} b^{7} {\left | b \right |} e^{8} + 2 \, A a^{3} b^{8} {\left | b \right |} e^{8}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (7 \, B b^{12} d^{5} {\left | b \right |} e^{3} - 33 \, B a b^{11} d^{4} {\left | b \right |} e^{4} - 2 \, A b^{12} d^{4} {\left | b \right |} e^{4} + 62 \, B a^{2} b^{10} d^{3} {\left | b \right |} e^{5} + 8 \, A a b^{11} d^{3} {\left | b \right |} e^{5} - 58 \, B a^{3} b^{9} d^{2} {\left | b \right |} e^{6} - 12 \, A a^{2} b^{10} d^{2} {\left | b \right |} e^{6} + 27 \, B a^{4} b^{8} d {\left | b \right |} e^{7} + 8 \, A a^{3} b^{9} d {\left | b \right |} e^{7} - 5 \, B a^{5} b^{7} {\left | b \right |} e^{8} - 2 \, A a^{4} b^{8} {\left | b \right |} e^{8}\right )}}{b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

(7*B*b^2*d*abs(b) - 5*B*a*b*abs(b)*e - 2*A*b^2*abs(b)*e)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqr
t(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + 1/15*(((b*x + a)*(15*(B*b^9*d^2*abs(b)*e^6 - 2*B*a*b^8*d*abs(b)*e
^7 + B*a^2*b^7*abs(b)*e^8)*(b*x + a)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9) + 23*(7*B*b^10*d^3*abs(b)*e^5
- 19*B*a*b^9*d^2*abs(b)*e^6 - 2*A*b^10*d^2*abs(b)*e^6 + 17*B*a^2*b^8*d*abs(b)*e^7 + 4*A*a*b^9*d*abs(b)*e^7 -
5*B*a^3*b^7*abs(b)*e^8 - 2*A*a^2*b^8*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9)) + 35*(7*B*b^11*d
^4*abs(b)*e^4 - 26*B*a*b^10*d^3*abs(b)*e^5 - 2*A*b^11*d^3*abs(b)*e^5 + 36*B*a^2*b^9*d^2*abs(b)*e^6 + 6*A*a*b^1
0*d^2*abs(b)*e^6 - 22*B*a^3*b^8*d*abs(b)*e^7 - 6*A*a^2*b^9*d*abs(b)*e^7 + 5*B*a^4*b^7*abs(b)*e^8 + 2*A*a^3*b^8
*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9))*(b*x + a) + 15*(7*B*b^12*d^5*abs(b)*e^3 - 33*B*a*b^1
1*d^4*abs(b)*e^4 - 2*A*b^12*d^4*abs(b)*e^4 + 62*B*a^2*b^10*d^3*abs(b)*e^5 + 8*A*a*b^11*d^3*abs(b)*e^5 - 58*B*a
^3*b^9*d^2*abs(b)*e^6 - 12*A*a^2*b^10*d^2*abs(b)*e^6 + 27*B*a^4*b^8*d*abs(b)*e^7 + 8*A*a^3*b^9*d*abs(b)*e^7 -
5*B*a^5*b^7*abs(b)*e^8 - 2*A*a^4*b^8*abs(b)*e^8)/(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a^2*b^4*e^9))*sqrt(b*x + a)/(b
^2*d + (b*x + a)*b*e - a*b*e)^(5/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(7/2),x)

[Out]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(7/2), x)

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